We look at the property of being low for isomorphism, restricted to certain classes of structures – if C is a class of structures, a set D is low for C isomorphism iff for any two structures in C, if D computes an isomorphism between them then there is a computable isomorphism between them. In particular we will show that exactly those sets which cannot compute non-zero Δ₂ degrees are low for ω-isomorphism (when ω is viewed purely as an order), and we will show that no set which computes a non-zero Δ₂ set or which computes a separating set for any two computably inseparable c.e. degrees is low for equivalence class isomorphism.